We are given positive integer numbers from $1$ to $10$, and we have to pick $3$ numbers from those $10$ so that the sum of those numbers (repetition of numbers in a sum is not allowed):
$a)$ equals $9$
$b)$ is less then $9$
$a)$ It is clear that those sums are: $ \quad6+2+1=9;\quad 5+3+1=9; \quad4+3+2=9$ . However, I am interested in method or explicit formula for solving this type of problem, do I have to partition the integers by cases, which could be a long process. For example, what if the sum of that $3$ numbers from some given set was supposed to be $857$ instead of $9$, the partitioning of $857$ could last very long.
$b)$ The same slow method of partitioning by cases comes to mind, of course, excluding the integers $10,9,8,7,6$.